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In particular, the RMT does not lead to iteration cycles on the notorious example by Ascher and Osborne [7] in contrast to the NMT. 6 Natural Monotonicity for LISA-Newton methods In this section we give a detailed description of an afﬁne covariant globalization strategy for a Newton-type method based on iterative linear algebra. The linear solver must supply error estimates in the variable space. , Saad [135]) and GBIT (due to Deuﬂhard et al. [45]). 2. 3) which we present in Chapter 6. 1: Rosenbrock-type example (adapted from Bock [24]): The Newton increment (solid arrow) is almost perpendicular to the direction of steepest descent for the classical level function T (z|I) (dashed arrow) in the initial iterate (marked by ◦).

4 Elements of optimization theory In this short chapter we consider the NLP minimize n x∈R s. t. 1c) → Rm are twice continuously differentiable functions where f and the sets E and I form a partition of {1, . . , m} =: m = E ∪˙ I . 1) is called Equality Constrained Optimization Problem (ECOP). 1 Basic deﬁnitions We follow Nocedal and Wright [121] in the presentation of the following basic deﬁnitions. 1. The set F = {x ∈ Rn | gi (x) = 0, i ∈ E , gi (x) ≥ 0, i ∈ I } is called feasible set. 2. A point x ∈ F is called feasible point.